Oscillatory disintegration of a trans-Alfvénic shock: A magnetohydrodynamic simulation

It has long been believed that trans-Alfvénic shock waves (TASWs), at which the flow velocity passes over the Alfvén velocity, cannot exist in the real world. Since a stationary trans-Alfvénic shock transition was obtained in a numerical simulation [1], this conventional view point was replaced by an opposite view point. The overall claim was that there is no principal difference between TASWs and fast and slow shocks, at which the flow is superand sub-Alfvénic, respectively. At the same time, the contradiction inherent in a stationary TASW, which follows from an analytical theory, was not lifted. To reconcile this contradiction, it was suggested that a TASW exists in an unsteady state in which it is repeatedly destroyed and recovered [2]. In the present paper, we show by way of magnetohydrodynamic (MHD) simulation that the evolution of a TASW may have the form of oscillatory disintegration, i.e., reversible transformation into another TASW. The disintegration of an arbitrary hydrodynamic discontinuity was considered for the first time by Kotchine [3]. After that, Bethe [4] studied the disintegration of shock waves. In the absence of a magnetic field, the shock may disintegrate only in a medium with anomalous thermodynamic properties. The magnetic field enlarges the number of possible discontinuous structures thus giving additional degrees of freedom for the disintegration. The disintegration configurations of arbitrary MHD discontinuities were obtained in Refs. [5]. Furthermore, it has been shown that trans-Alfvénic shock transitions can be realized also through a set of several discontinuities [6], in contrast with fast and slow transitions. However, this fact on its own does not assure that the shock disintegrates. The important feature that predetermines the disintegration of TASWs is their nonevolutionarity. The problem of evolutionarity was initially formulated for the fronts of combustion [7] and hydrodynamic discontinuities [8]. Evolutionarity is a property of a discontinuous flow to evolve in such a way that the flow variation remains small under the action of a small perturbation. This is not the case for a nonevolutionary discontinuity. At such a discontinuity, the system of boundary conditions, which follow from the conservation laws, does not have the unique solution for the amplitudes of outgoing waves generated by given incident waves. From a mathematical view point this means that the number of unknown parameters (the amplitudes of the outgoing waves and the discontinuity displacement) is incompatible with the number of independent equations. Since a physical problem must have the unique solution, the assumption that the perturbation of a nonevolutionary discontinuity is infinitesimal leads to a contradiction. In fact, the infinitesimal perturbation results in disintegration, i.e., finite variation of the initial flow, or transformation into some other unsteady configuration. The evolutionarity requirement gives additional restrictions on the flow parameters at a shock, compared to the condition of the entropy increase. The restrictions appear because the direction of wave propagation (toward the discontinuity surface or away from it), and thus the number of the outgoing waves, depends on the flow velocity. If the velocity is large enough then the given wave may be carried down by the flow. Therefore, at an evolutionary discontinuity, the flow velocity must be such that it provides the compatibility of the boundary equations. This form of evolutionarity condition was applied to MHD shock waves in Refs. [9,10]. As a result, the fast and slow shocks are evolutionary, while the TASWs are